Existence of extremizers for self-volume in higher dimensions

Establish whether, for each dimension n > 2, there exists a centrally symmetric convex body B ⊂ R^n that minimizes or maximizes the self-volume V_B^{(n)} defined by V_B^{(n)} := P(B)/n with self-perimeter P(B) = ∫_{∂*B} [ V(B^{(n−1)}(ν_x)) / 𝓗_{n−1}(B^{(n−1)}(ν_x)) ] d𝓗_{n−1}(x).

Background

After defining self-volume through a recursive boundary integral and proving invariance properties, the authors raise the question of extremizers in higher dimensions. While several two-dimensional extremal results exist for self-perimeter, the analogous existence of shapes that minimize or maximize V_B^{(n)} in higher dimensions is not established.

The problem seeks to ascertain whether extremal centrally symmetric convex bodies exist for the self-volume functional in dimensions n > 2, thereby initiating a program analogous to classical extremal shape problems in convex geometry.